FRANKFURT—SAP SE expects its chief executive, Bill McDermott, to return to the software maker’s headquarters in Germany early next month after he lost an eye in an accident, a spokesman said Wednesday.

Mr. McDermott fell down stairs in early July at his brother’s home in the U.S. and a water glass he was holding hit his left eye, according to the company. Doctors weren’t able to save the eye. The accident happened following a birthday party for Mr. McDermott’s father. Spokesman Daniel Reinhardt declined to disclose the location...